In this paper, we obtain a complete list of inequivalent irreducible representations of the compact quantum group Uq(2) for nonzero complex deformation parameters q, which are not roots of unity. The matrix coefficients of these representations are described in terms of the little q-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of L2(U q(2)) is obtained. Thus, we have an explicit description of the Peter–Weyl decomposition of Uq(2). As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum group Uq(2).

中文翻译：

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复杂变形参数的紧致量子群 Uq(2) 的表示和分类

在本文中，我们获得了紧致量子群的不等价不可约表示的完整列表üq(2)对于非零复杂变形参数q，它们不是统一的根。这些表示的矩阵系数用小q-雅可比多项式。Haar 状态被证明是忠实的，并且是大号2(ü q(2))获得。因此，我们有一个明确的描述 Peter-Weyl 分解üq(2). 作为一个应用程序，我们讨论傅里叶变换并建立 Plancherel 公式。我们还描述了将两个不可约表示的张量积分解为不可约分量。最后，我们对紧致量子群进行分类üq(2).